Quick Review of

Expanding Determinants

A matrix is a collection of numbers -- or other items -- in a rectangular row-and-column format. A matrix might look like this:


The numbers or matrix elements are usually enclosed with large parentheses (which I have trouble drawing with my tools) or in square brackets (as I have shown here). That's all a matrix is -- just a collection of elements in a row-and-column format.

However, a determinant looks somewhat like a matrix but has a numerical value. The elements of a determinant are enclosed between vertical bars and might look like this:


While this looks somewhat like a matrix it can be evaluated -- it has a particular numerical value. There are various ways to calculate this value or to expand the determinant. One of the ways is known as an expansion by cofactors. Go across one row (or column) and multiply each element with an alternating sign and a smaller determinant of the remaining elements when the elements of that row and column are removed. What in the world does that mean????? Let's expand this determinant above as an example,

Finally, when we get to two-by-two determinants, we can fairly easily evaluate those:

Throughout this expansion and evaluation, watch the signs!

Now we can apply this to evaluating the cross product or writing the cross product as a determinant:

Vector Product (2)

Return to ToC, Ch11, Rolling Motion

(c) Doug Davis, 2001; all rights reserved